
ERTS Algorithm:
We design ERTS, an algorithm that is similar to the structure of CVaR-TS in [1]} but using entropic risk instead of CVaR as the risk measure. This algorithm uses Thompson sampling [2]} as explored for mean-variance bandits in [3]} and CVaR bandits in [1]}.

Comprehensive regret bounds:
We provide theoretical analysis of the ERTS algorithm for Gaussian bandits with bounded variances. We state and prove both upper and lower bounds, showing that ERTS is the first asymptotically optimal algorithm that solves the entropic risk multi-armed bandit problem. Our proof techniques solidify the novel \(\xi \) -trick in [1]}, and affirm future analysis on MABs involving generalised risk measures.

